CN117075635B - Multi-spacecraft control method based on trigger strategy - Google Patents
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Abstract
The invention discloses a multi-spacecraft control method based on a trigger strategy, which comprises the following steps: step 1, establishing a spacecraft kinematics and dynamics model by using a quaternion method; step 2, taking interference and uncertainty factors into consideration, and establishing a spacecraft tracking model on the basis of the step 1; step 3, considering the existing fixed time convergence sliding mode surface, and designing a sliding mode variable s by combining the quaternion spacecraft mathematical model established in the step 1; step 4, designing a dynamic event trigger mechanism; step 5, designing a fixed time convergence observer; and 6, designing a spacecraft formation fixed time stable control law and a self-adaptive law based on an event trigger mechanism on the basis of completion of the steps 1-5, and realizing tracking of the spacecraft formation on the expected gesture. The controller designed by the invention can reduce the communication times between spacecraft formations.
Description
Technical Field
The invention belongs to the field of nonlinear system control, relates to a spacecraft attitude control algorithm, and in particular relates to a multi-spacecraft control method based on a trigger strategy.
Background
In a multi-spacecraft formation system, the number of the spacecraft is large, and unified instructions are required to be obeyed, so that the requirement on a communication network is high. But individual spacecraft in a spacecraft formation are typically smaller in size for cost savings. Therefore, the power of the communication device that can be mounted is limited, and the communication capability is insufficient.
In spacecraft formation control, a controller using a trigger mechanism may be considered to reduce the number of communications in spacecraft formation. From the aspect of stable precision, the controller can be designed by utilizing a sliding mode method. The spacecraft formation tracking observer is controlled to observe the desired pose using a control algorithm that is stable in real fixed time. In addition, in consideration of factors such as interference and uncertainty, robustness needs to be improved by using methods such as self-adaption.
Disclosure of Invention
The invention aims to provide a multi-spacecraft control method based on a trigger strategy, which adopts a spacecraft dynamics and kinematics model represented by quaternions, considers external interference and uncertainty, uses an adaptive method to process, selects a sliding mode surface capable of converging at fixed time, designs a fixed time stable posture observer to observe expected postures, and further designs a fixed time stable algorithm to realize tracking of expected signals by each spacecraft in a spacecraft formation.
The invention aims at realizing the following technical scheme:
A multi-spacecraft control method based on a trigger strategy comprises the following steps:
Step 1, establishing a spacecraft kinematics and dynamics model by using a quaternion method, wherein the specific form is as follows:
Wherein J i is the moment of inertia of the ith spacecraft, omega i is the rotation angular velocity signal of the ith spacecraft, Is a square matrix derived from omega i, q i and q i0 are attitude signals of an ith spacecraft represented by a quaternion representation method, u i is a control signal of the ith spacecraft, w i is external interference moment of the ith spacecraft, and I 3 is a 3-order identity matrix;
step 2, taking interference and uncertainty factors into consideration, and establishing a spacecraft tracking model based on the step 1, wherein the specific form is as follows:
Wherein J i0 is the nominal moment of inertia of the ith spacecraft, omega ei is the error value of the gesture information and the expected gesture of the ith spacecraft, omega d is the expected gesture value of spacecraft formation, q ei and q 0,ei are error gesture signals of the ith spacecraft represented by a quaternion representation, and R (q ei) is a rotation matrix;
Step 3, considering the existing fixed time convergence sliding mode surface, and combining the quaternion spacecraft mathematical model established in the step 1, designing a sliding mode variable s i, wherein the specific form is as follows:
si＝ωei+sgn(q0,ei(0))K1Saui
Wherein, l 1、l2、η、p1、K1 is a fixed parameter;
step 4, designing a dynamic event triggering mechanism as follows:
Wherein e i is the event trigger error, Representing the kth triggering moment of the ith spacecraft, wherein eta is a constant positive variable;
Step 5, designing a fixed time convergence observer as follows:
Wherein, Representing an estimate of the desired information for the ith spacecraft,Gamma 1,γ2 is a positive odd number, gamma 1<γ2,aij、bi、di is a spacecraft formation parameter, sig (x) is a special function, and ：sigα(x)＝[sgn(x1)·|x1|α sgn*(x1)·|x1|α sgn(x1)·|x1|α]T,sgn is a sign function assuming x= [ x 1 x2 x3]T;
Step 6, designing a spacecraft formation fixed time stable control law and a self-adaptive law based on an event trigger mechanism on the basis of completion of the steps 1-5, and tracking expected gestures by the spacecraft formation, wherein the spacecraft formation fixed time stable control law and the self-adaptive law based on the event trigger mechanism are as follows:
Wherein,
Π＝αdiag{|ωei|α-1}，
S i is a sliding mode variable of the ith spacecraft, s j(tk) is sliding mode information of the jth spacecraft obtained at time T k, k 1、k2、μ、λ、α、β、z1、z2 is a constant, and t=1+|ω ei||+||ωei||2.
Compared with the prior art, the invention has the following advantages:
1. the posture consistency of the multiple spacecrafts needs to be adjusted through the interaction among the spacecrafts, so that the state that the postures of the spacecrafts are consistent is achieved. This problem has been an important issue in spacecraft formation flight, and has attracted a great deal of attention. The invention designs the fixed time controller based on the event triggering mechanism, so that spacecraft formation can reach a state with consistent gestures.
2. In spacecraft formation, to save costs, the individual spacecraft is smaller in size and fewer communication elements can be loaded, which results in less communication burden that the spacecraft can bear. However, the designed spacecraft formation attitude consistency controller requires a large amount of communication among the spacecraft. The controller designed by the invention can reduce the communication times between spacecraft formations.
Drawings
FIG. 1 is a system block diagram;
FIG. 2 is a graph of the number of dynamic event triggers for spacecraft 1;
FIG. 3 is a graph of the number of dynamic event triggers for spacecraft 2;
FIG. 4 is a graph of the number of dynamic event triggers for spacecraft 3;
FIG. 5 is a graph of the number of dynamic event triggers for spacecraft 4;
Fig. 6 is an observation error of the observer 1;
Fig. 7 is an observation error of the observer 2;
fig. 8 is an observation error of the observer 3;
Fig. 9 is an observation error of the observer 4;
fig. 10 is an angular velocity and attitude of the spacecraft 1;
FIG. 11 is an angular velocity and attitude of the spacecraft 2;
fig. 12 is an angular velocity and attitude of the spacecraft 3;
Fig. 13 is an angular velocity and attitude of the spacecraft 4;
Detailed Description
The following description of the present invention is provided with reference to the accompanying drawings, but is not limited to the following description, and any modifications or equivalent substitutions of the present invention should be included in the scope of the present invention without departing from the spirit and scope of the present invention.
The invention provides a multi-spacecraft control method based on a trigger strategy, which adopts a quaternion modeling method of a spacecraft, considers the uncertainty of model parameters and external environment interference of the spacecraft, designs a gesture observer based on an event trigger mechanism and a multi-spacecraft formation gesture control algorithm, and realizes gesture tracking control of spacecraft formation. The method specifically comprises the following steps:
and step 1, establishing a spacecraft kinematics and dynamics model.
Considering that the quaternion representation method of the gesture has the characteristics of no singular, simple calculation and the like, the quaternion method is selected to be used for establishing a spacecraft kinematics and dynamics model, and the specific form is as follows:
Wherein ω i is a rotation angular velocity signal of the ith spacecraft, q i and q i0 are attitude signals of the ith spacecraft represented by a quaternion representation method, u i is a control signal of the ith spacecraft, J i is a moment of inertia of the ith spacecraft, and w i is an external disturbance moment of the ith spacecraft.
A quaternion is an extension of a complex number, and consists of a real part and an imaginary part, which is usually represented by three imaginary units i, j, k, i.e. the quaternion can be written in the form a+ b i + c j + d k, where a, b, c, d is a real number. q i and q i0 are vector and scalar parts of unit quaternions, and q i and q i0 satisfyI 3 is a 3-order identity matrix.Is a square matrix derived from omega i, satisfying the following conditions: Here the dimension ω i is 3, written as:
step 2, taking interference and uncertainty factors into consideration, and establishing a spacecraft tracking model based on the step 1, wherein the specific form is as follows:
Wherein J i0 is the nominal moment of inertia of the ith spacecraft, omega ei＝ωi-R(qei)ωd,ωei is the error value of the pose information and the expected pose of the ith spacecraft, omega d is the expected pose value of spacecraft formation, Like ω ×, the square derived for ω ei, q ei and q 0,ei are the error attitude signals of the ith spacecraft represented by the quaternion representation, and R (q ei) is the rotation matrix, defined as:
Considering the influence of uncertainty factors of the spacecraft system, the inertia matrix J i consists of a nominal inertia matrix J i0 and an uncertainty part DeltaJ i, namely J i＝Ji0+ΔJi. Lumped interference can be written as: To facilitate subsequent analysis, reasonable assumptions are made about the spacecraft systems described above:
Suppose 1: the perturbation is assumed to be bounded, i.e., ||w i||≤wmax,wmax > 0 and is a known constant.
Suppose 2: assuming that the spacecraft rotational angular velocity omega i is bounded, i.e., ||ω i||≤δ1,δ1 > 0 and is a known constant.
Suppose 3: assuming rotational angular acceleration of spacecraftBounded, i.eDelta 2 > 0 and is a known constant.
Step 3, considering the existing fixed time convergence sliding mode surface, and combining the quaternion spacecraft mathematical model established in the step 1, designing a sliding mode variable s i, wherein the specific form is as follows:
si＝ωei+sgn(q0,ei(0))K1Saui
Where ω ei is the angular velocity error in the quaternion notation, q ei is the attitude error in the quaternion notation, and l 1、l2、η、p1、K1 is a fixed parameter. For a designed slip-form surface, it can be demonstrated that the slip-form surface eventually can converge in a fixed time.
Taking outWhen S i =0, there are:
ωei＝-sgn(q0,ei)K1Saui
Thus, it is possible to obtain:
q0,ei(0)≥0
So that:
the system settling time is stable as seen by the following quotients:
and (5) lemma: if the system has Lyapunov equation, the equation is as follows: and alpha and m are positive real numbers, 0< m <1, the system fixed time converges, and the convergence time is T=1/(alpha m) +2/[ pi alpha (1-m) ].
Therefore, the designed slip-form surface can converge in a fixed time.
Step 4, designing a dynamic event trigger mechanism, wherein the communication loss in the formation of the spacecraft can be greatly reduced by using the trigger mechanism, and the dynamic event trigger mechanism is designed as follows:
Wherein e i is the event trigger error, taken as Representing the kth trigger time of the ith spacecraft,May represent a series of trigger time points for the ith spacecraft, η being a constant positive variable. The purpose of such design is to enable the spacecraft error signal to update the signal when exceeding a certain interval, thereby avoiding the spacecraft to update the signal from time to time.
The gano phenomenon (Zeno behavior) refers to infinite triggering in a limited time during event triggering control, and for a designed event triggering condition, the triggering condition is continuously satisfied, and the controller cannot effectively adjust the triggering. This behavior is therefore to be avoided in the design of trigger conditions, and the following is to be established during the analysis proof process:
Where δ is a positive constant. That is, any two trigger moments cannot be equal. This avoids multiple triggers at a certain time. The control method designed by the invention excludes zeno phenomenon as follows:
||ei||-k||si||≤η
k||si||-||ei||≥-η
η≥0
Furthermore:
b s must be present between t k i and t such that
And because ofSo there is
Step 5, considering that not all spacecrafts can acquire the desired information in the formation of the spacecrafts, the observer observation signals are designed. The following fixed time convergence observer was designed:
Wherein, Representing an estimate of the desired information for the ith spacecraft,Gamma 1,γ2 is a positive odd number, gamma 1<γ2,aij、bi、di is a spacecraft formation parameter, q i is a spacecraft formation desired pose, i.e. q d,Is a variable in the observer. sig (x) is a special function, and ：sigα(x)＝[sgn(x1)·|x1|α sgn(x1)·|x1|α sgn(x1)·|x1|α]T,sgn is a sign function assuming that x= [ x 1 x2 x3]T ].
Assuming that the communication capacity between the spacecrafts is denoted by a ij, a ij =1 is denoted by a ij =1, and if a ij =0, no communication capacity is denoted between the ith and the jth spacecrafts. And communication among the spacecrafts is undirected, namely a ij＝aji, wherein the matrix A is used for representing the communication capacity of formation of the spacecrafts, and the matrix A is a symmetrical matrix. In addition, there areB i is used to measure the ability of the ith spacecraft to acquire the desired attitude information. If the ith spacecraft can acquire the desired information, b i =1, otherwise b i =0.
To demonstrate observer stability, the following Lyapunov equation may be taken:
derivation of V 0:
For γ in the design of the invention, there is x γsigγ(x)＝x2γ. Taking out The method can obtain:
Thus, the designed observer fixed time converges.
And 6, designing a spacecraft formation fixed time stable control law and a self-adaptive law based on an event trigger mechanism on the basis of completion of the steps 1-5, and realizing tracking of the spacecraft formation on the expected gesture. The spacecraft formation fixed time stable control law and the self-adaptive law based on the event triggering mechanism are as follows:
For the convenience of calculation, pi and L are taken from the spacecraft model in the step 2 as follows:
Π＝αdiag{|ωei|α-1}，
s i is the sliding mode variable of the ith spacecraft, s j(tk) is the sliding mode information of the jth spacecraft obtained at the time t k. k 1、k2、μ、λ、α、β、z1、z2 is a constant. T=1+||ω ei||+||ωei||2.
The following Lyapunov equation was chosen:
V＝V1+V2
wherein:
Deriving V 1, we can obtain:
the control law proposed above is brought in to obtain:
Assuming a=k 1sigm(si),b＝k2sign(si), then- (k 1sigm(si)+k2sign(si))d) can be denoted as- (a+b) d. Then:
So there are:
Wherein, kappa 1、κ2、κ3 is a positive number, so:
So that it is possible to obtain:
As will be seen from the following quotation, the control law stabilizes the system for a virtually fixed time.
For nonlinear systemsF (0) =0, x e R n, selecting a suitable Lyapunov equation V, if V satisfies the following equation:
Wherein alpha, beta, p, q, k.epsilon.R +, pk < 1, gk > 1,0 < θ < ++ infinity, the system balance point x=0 is stable for a virtually fixed time. The convergence time satisfies: Wherein,
Assuming that the number of spacecraft in spacecraft formation is 4, part of parameters are as follows:
Rotational inertia of spacecraft:
Controller parameters:
α＝1.1,β＝diag(0.250.250.25),λ＝diag(2.492.492.49),γ＝0.5,z1＝0.0001,z2＝0.1,k1＝20,k2＝20
Initial state of spacecraft:
q1＝[0 -0.1 0.2]T,q10＝0.9747,ω1＝[0.1 0.5 -0.1]T
q2＝[0.1 0 0.1]T,q20＝0.9899,ω2＝[0.7 0.5 -0.5]T
q3＝[0 0.1 -0.1]T,q30＝0.9899,ω3＝[0.6 -0.3 -0.8]T
q4＝[-0.1 0 -0.2]T,q40＝0.9747,ω4＝[0.2 0.6 0.1]T
time-varying interference received by the spacecraft:
d1＝[0.005*sin(t) 0.008*cos(t) -0.002*sin(t)]T
d2＝[0.002 0.004*sin(t) -0.002*cos(t)]T
d3＝[0.005*sin(t) -0.008 -0.002*cos(t)]T
d4＝[0 0 -0.005*sin(t)]T
the simulation results are shown in fig. 2-13, and compared with a control algorithm stable in a limited time, the algorithm can reach stability in a faster time and has better effect.
Claims (3)
1. The multi-spacecraft control method based on the trigger strategy is characterized by comprising the following steps of:
Step 1, establishing a spacecraft kinematics and dynamics model by using a quaternion method, wherein the specific form is as follows:
Wherein J i is the moment of inertia of the ith spacecraft, omega i is the rotation angular velocity signal of the ith spacecraft, Is a square matrix derived from omega i, q i and q i0 are attitude signals of an ith spacecraft represented by a quaternion representation method, u i is a control signal of the ith spacecraft, w i is external interference moment of the ith spacecraft, and I 3 is a 3-order identity matrix;
step 2, taking interference and uncertainty factors into consideration, and establishing a spacecraft tracking model based on the step 1, wherein the specific form is as follows:
Wherein J i0 is the nominal moment of inertia of the ith spacecraft, omega ei is the error value of the gesture information and the expected gesture of the ith spacecraft, omega d is the expected gesture value of spacecraft formation, q ei and q 0,ei are error gesture signals of the ith spacecraft represented by a quaternion representation, and R (q ei) is a rotation matrix;
Step 3, considering the existing fixed time convergence sliding mode surface, and combining the quaternion spacecraft mathematical model established in the step 1, designing a sliding mode variable s i, wherein the specific form is as follows:
si＝ωei+sgn(q0,ei(0))K1Saui
Wherein, l 1、l2、η、p1、K1 is a fixed parameter;
step 4, designing a dynamic event triggering mechanism as follows:
Wherein e i is the event trigger error, Representing the kth triggering moment of the ith spacecraft, wherein eta is a constant positive variable;
Step 5, designing a fixed time convergence observer as follows:
Wherein, Representing an estimate of the desired information for the ith spacecraft,Gamma 1,γ2 is a positive odd number, gamma 1<γ2,aij、bi、di is a spacecraft formation parameter, sig (x) is a special function, and ：sigα(x)＝[sgn(x1)·|x1|α sgn(x1)·|x1|α sgn(x1)·|x1|α]T,sgn is a sign function assuming x= [ x 1 x2 x3]T;
Step 6, designing a spacecraft formation fixed time stable control law and a self-adaptive law based on an event trigger mechanism on the basis of completion of the steps 1-5, and tracking expected gestures by the spacecraft formation, wherein the spacecraft formation fixed time stable control law and the self-adaptive law based on the event trigger mechanism are as follows:
Wherein,
Π＝αdiag{|ωei|α-1}，
S i is a sliding mode variable of the ith spacecraft, s j(tk) is sliding mode information of the jth spacecraft obtained at time T k, k 1、k2、μ、λ、α、β、z1、z2 is a constant, and t=1+|ω ei||+||ωei||2.
2. The method of multi-spacecraft control based on trigger strategy according to claim 1, characterized in that R (q ei) is defined as:
3. the method for controlling a multi-spacecraft based on a trigger strategy according to claim 1, characterized in that said method comprises the steps of
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